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Theorems are given concerning the order (i.e., rate) of convergence of a successive interpolation process for finding simple zeros of a function or its derivatives, using only function evaluations. Special cases include the successive linear interpolation process for finding zeros, and a parabolic interpolation process for finding turning points. Results on interpolation and finite differences include weakening the hypotheses of a theorem of Ralston on the derivative of the error in Lagrangian interpolation. The theoretical results are applied to given algorithms for finding zeros or local minima of functions of one variable, in the presence of rounding errors. The algorithms are guaranteed to converge nearly as fast as would bisection or Fibonacci search, and in most practical cases convergence is superlinear, and much faster than for bisection or Fibonacci search. (Author).
DIVOutstanding text for graduate students and research workers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. /div
We discuss the secant method:where are initial guesses. In the Regula Falsi variation we start with initial guesses and such that ; after an iteration similar to the above we replace either a or b by the new value depending on which of or has the same sign as . Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. We discuss convergence and efficiency of most of the methods considered. We treat methods involving quadratic of higher order interpolation and rational approximation. We also discuss the bisection method where again and we set . We replace a or b by c according to the sign of as in the Regula Falsi method. Various generalizations are described, including some for complex roots. Finally we consider hybrid methods involving two or more of the previously described methods.
Encompassing all the major topics students will encounter in courses on the subject, the authors teach both the underlying mathematical foundations and how these ideas are implemented in practice. They illustrate all the concepts with both worked examples and plenty of exercises, and, in addition, provide software so that students can try out numerical methods and so hone their skills in interpreting the results. As a result, this will make an ideal textbook for all those coming to the subject for the first time. Authors' note: A problem recently found with the software is due to a bug in Formula One, the third party commercial software package that was used for the development of the interface. It occurs when the date, currency, etc. format is set to a non-United States version. Please try setting your computer date/currency option to the United States option . The new version of Formula One, when ready, will be posted on WWW.
The purpose of this book is to introduce and explain research at the boundary between two fields that view problem solving from different perspectives. Researchers in operations research and artificial intelligence have traditionally remained separate in their activities. Recently, there has been an explosion of work at the border of the two fields, as members of both communities seek to leverage their activities and resolve problems that remain intractable to pure operations research or artificial intelligence techniques. This book presents representative results from this current flurry of activity and provides insights into promising directions for continued exploration. This book should be of special interest to researchers in artificial intelligence and operations research because it exposes a number of applications and techniques, which have benefited from the integration of problem solving strategies. Even researchers working on different applications or with different techniques can benefit from the descriptions contained here, because they provide insight into effective methods for combining approaches from the two fields. Additionally, researchers in both communities will find a wealth of pointers to challenging new problems and potential opportunities that exist at the interface between operations research and artificial intelligence. In addition to the obvious interest the book should have for members of the operations research and artificial intelligence communities, the papers here are also relevant to members of other research communities and development activities that can benefit from improvements to fundamental problem solving approaches.
George Dantzig is widely regarded as the founder of this subject with his invention of the simplex algorithm in the 1940's. In this second volume, the theory of the items discussed in the first volume is expanded to include such additional advanced topics as variants of the simplex method; interior point methods, GUB, decomposition, integer programming, and game theory. Graduate students in the fields of operations research, industrial engineering and applied mathematics will thus find this volume of particular interest.
A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data. This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.
The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. The collection of 21 papers is divided into five main areas: iterative methods for linear systems, solution of least squares problems, matrix factorizations and applications, orthogonal polynomials and quadrature, and eigenvalue problems. Commentaries for each area are provided by leading experts: Anne Greenbaum, Ake Bjorck, Nicholas Higham, Walter Gautschi, and G. W. (Pete) Stewart. Comments on each paper are also included by the original authors, providing the reader with historical information on how the paper came to be written and under what circumstances the collaboration was undertaken. Including a brief biography and facsimiles of the original papers, this text will be of great interest to students and researchers in numerical analysis and scientific computation.
The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. Including commentaries by leading experts and a brief biography, this text will be of great interest to students and researchers in numerical analysis and scientific computation.